Let $F$ be a field (or, more generally, a division ring). A vector space $V$ over $F$ is a set with two operations, $+: V \times V \longrightarrow V$ and $\cdot: F \times V \longrightarrow V$ , such that

1. $(\u+\v)+\w = \u+(\v+\w)$ for all $\u,\v,\w \in V$
2. $\u+\v=\v+\u$ for all $\u,\v\in V$
3. There exists an element $\0 \in V$ such that $\u+\0=\u$ for all $\u \in V$
4. For any $\u \in V$ , there exists an element $\v \in V$ such that $\u+\v=\0$
5. $a \cdot (b \cdot \u) = (a \cdot b) \cdot \u$ for all $a,b \in F$ and $\u \in V$
6. $1 \cdot \u = \u$ for all $\u \in V$
7. $a \cdot (\u+\v) = (a \cdot \u) + (a \cdot \v)$ for all $a \in F$ and $\u,\v \in V$
8. $(a+b) \cdot \u = (a \cdot \u) + (b \cdot \u)$ for all $a,b \in F$ and $\u \in V$

Equivalently, a vector space is a module $V$ over a ring $F$ which is a field (or, more generally, a division ring).

The elements of $V$ are called vectors, and the element $\0 \in V$ is called the zero vector of $V$ .